Calculate $12345 \pmod{11}$ and $12345\times 98765 \pmod{11}$ by hand.

Question

I know the answers to those questions, but I'd like to know how to do them by hand.

score 2 · Accepted Answer · answered Feb 14 '18 at 01:34

2

Hint:

$$10 \equiv -1 \pmod{11}$$

You might like to write $12345=10000+2000+300+40+5$.

answered Feb 14 '18 at 01:34

Siong Thye Goh

So getting that 10000(mod 11) is 1, 2000(mod 11) is 9, 300(mod 11) is 3, 40(mod 11) is 7, 5(mod 11) is 5, which is 25 altogether, and 25(mod 11) is 3? – tigerustin Feb 14 '18 at 01:38
yup, try to work with smaller number, we can write $9$ as $-2$ in mod $11$. Hence the question is just $1-2+3-4+5\equiv 3 \pmod{11}$ – Siong Thye Goh Feb 14 '18 at 01:40
I see, and you would do the same with $12345 \times 98765$? – tigerustin Feb 14 '18 at 01:41
Similar trick applies. note that $ab \pmod{c} \equiv (a \pmod{c}) (b \pmod{c}) $. Try to work with small numbers. don't multiply two $5$-digit numbers. – Siong Thye Goh Feb 14 '18 at 01:43

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